On nilpotent semigroups and solutions with finite stopping time (Q1269560)

The author deals with the equation \[ u'=Bu\tag{1} \] with a closed unbounded operator \(B\) defined on a dense domain of a separable Hilbert space \(H\). The basic problem is a description of conditions on \(B\) under which equation (1) possesses a nontrivial solution which becomes zero in finite time. The author presents a complete characterization of this phenomenon in terms of simple properties of \(B\), separately, in the partial case, when \(\text{Ker} B=\{0\}\), and in the general case. This result makes transpartent the well-known Hille-Phillips example of this phenomenon and generalizes some results presented by [\textit{A. M. Sinclair}, Continuous semigroups in Banach algebras, Cambridge University Press, UK (1982; Zbl 0493.46042)].